1. Modified gravity: Brief summary
Dec
GR100,
EWHA Woman’s University
Modified gravity: Brief summary
*Hyeong-Chan Kim
Korea National University of Transportation

2. There are diverse number of modified theories.
Introduction
Most modified theories which does not contain GR in a limit, dead already. (80~)
There are diverse number of modified theories.
Variances:
Direct modifications: Cartan, Brans-Dicke, Rosen bimetric…
Quantized gravity: loop quantum gravity, Noncommutative geometry, Horava-Lifschitz,
Unification with other forces: Kaluza-Klein, Supergravity
Do everything: M-theory ...
Deals direct modifications only.

3. Motivations for modified gravity
General relativity (GR) is quite successful. However,
Quantization of gravity (Noncommutative geometry, String theory, Horava-Lifschitz, … )
Low energy effective theories of (would be) quantum gravity are represented by the modified gravity theories.
Higher curvuture terms as a quantum gravitational corrections. (f(R ), Gauss-Bonnet, Lovelock, Born-Infeld… )
Spacetime singularities indicate that the theory is incomplete. (Einstein-Cartan, Eddington-ispired Born-Infeld)
Add other aspects e.g., spin (Einstein-Cartan), additional DoF (Brans-Dicke, Kaluza-Klein, … )
There exist phenomena which cannot be explained in GR, e.g, Dark energy (DGP) and Dark matter (MOND).

4. General Relativity Levi-Civita Connection Metric Curvature
= matter
Distance between two points
Gravity or Acceleration at a given coordinates system:
Determine the motion of particles.
Matter determines that the spacetime how to curve.

5. Modification 1) Add new DoF
Modify the RHS so that include new DoF: Scalar-tensor theory, Brans-Dicke, Tensor-Vector-Scalar theory,…
EoM: add stress-tensors for new fields
GR appears when the new DoFs are killed.
Spacetime singularity happens when the density diverges. (There may exist other singularities).
Same difficulties in quantization and singularities as GR.

6. Scalar-tensor theories
Dimensionally reduced effective theories of KK, string theories.
Jordan frame:
Conformal transformation:
Einstein frame:
Most general with 2nd order EoM: Horndeski’s theory

7. Brans-Dicke Theory of gravity (1961)
Motivation: Mach’s principle (The reference frame comes from the distribution of matters in the universe.)
In GR, geometry is determined by mass distribution.
However, it is not unique up to boundary condition.
Set the Newton constant be dynamical (1/G  ).
: general relativity
Present experimental limit:
Deviation from GR is tiny if exists.

8. General Relativity Levi-Civita Connection Metric Curvature
= matter
Distance between two points
Gravity or Acceleration at a given coordinates system:
Determine the motion of particles.
Matter determines that the spacetime how to curve.

9. Modification 2) Add higher curvatures
Modify the LHS so that include higher curvature terms: f(R), conformal gravity, Gauss-Bonnet gravity, Lovelock gravity, etc.
EoM:
Add higher curvature terms.
GR appears in the low curvature limit.
Spacetime singularity happens when the density diverges.
There exists other singularities.

10. Modification 2) f(R) theory
Generally covariant, Lorentz invariant.
Fourth order equation of motion
GR +massless scalar field with non-metric coupling with matter.
3 massless graviton= 2 tensor+1scalar.
Conformal transform:
Present bound: | a2| < 4 x 10-9 m2 = 2.3x 10-22GeV-2 (fifth force measurement)

11. General Relativity Levi-Civita Connection Metric Curvature
= matter
Distance between two points
Gravity or Acceleration at a given coordinates system:
Determine the motion of particles.
Matter determines that the spacetime how to curve.

12. Modification 3) Change theory structure
EiBI, Bi-gravity and other Palatini-f(R) gravity.
Metric
Affine -Connection
Curvature
Affine connection cannot be obtained from metric straight forwardly.
Matter
EiBI, Bi-gravity and other Palatini-(connection) based modified gravity: is not the spacetime curvature but the connection curvature.
1) GR happens in the zero matter limit.
2) Spacetime singularity may or may not appear even if the energy density diverges.

13. | |<3 x 105 m5s-2/kg (Casanellas,Pani,Lopes,Cardoso,2011)
Born-Infeld Generalization of Eddington gravity, inequivalent to GR:
Action ( )
is dependent on the connection only.
The matter field couples with the metric only.
Small = GR limit; Large = Eddington limit.
is a dimensionless constant related with the cosmological constant.
For vacuum, it is the same as GR. Inside matters, it deviates from GR.
| |<3 x 105 m5s-2/kg (Casanellas,Pani,Lopes,Cardoso,2011)
EiBI gravity(Banados, Ferreira; 2010)

14. The equation of motions
Define auxiliary metric:
Metric
Affine -Connection
Curvature
Matter
Now, the variation of the action w.r.t gives, where,
The metric compatibility determines,
Variation with respect to the metric,

15. General Relativity Levi-Civita Connection Metric Curvature
= matter
Distance between two points
Gravity or Acceleration at a given coordinates system:
Determine the motion of particles.
Matter determines that the spacetime how to curve.

16. Modification 4) Einstein Cartan theory
The connection has non-zero torsion.
Ricci tensor is asymmetric.
Spinning particles feels the torsion.
contorsion tensor
The autoparallels are not necessarily the longest (shortest) lines.
On the other hand, minimizing proper length:

17. Modification 4) Einstein Cartan theory
EC emerges as the minimal description of spacetime of local gauge theory of Poincare group.
Non-renormalizable (spin interaction ~ 4-fermion interaction,
torsion has vanishing canonical momentum)
The theory is different from GR at very high spin-densities of matter.
For electron,
Cosmological singularity can be avoided only under unrealistic circumstances.

18. theories not categorized
There are many other theories that does not belong to these categories.
Gauge gravity theory
Chern-Simon theory in 2+1 dim
Theories breaking diffeomorphism invariance
Breaks Lorentz invariance: Horava-Lifschitz, Einstein Ether,…
Massive gravity
Extension to Higher dimensions

19. Testing modified theories
Self-consistency: tachyon, ghost, higher order poles.
Completeness:
A theory of gravity must be capable of analyzing the outcome of every experiment of interest.
Classical tests:
gravitational redshift
Gravitational lensing
Anomalous perihelion advances of the planets
Agreement with Newtonian mechanics and special relativity
The Einstein equivalence principle
Parametric post-Newtonian formalism(weak field)
Strong gravity and gravitational waves (BH horizon, speed of GW, etc.)
Cosmological tests (Dark matter, galaxy rotation curve, Tully-Fisher relation), gravitational lensing due to galactic cluster, CMB, (Dark energy: Supernova brightness )

20. arXiv:1501.07274, Testing GR with present and future astrophysical observations

21. Thanks for leastening!

22. Fourth order gravity (second order of Riemann)
Metric theories
Fourth order gravity (second order of Riemann)
f(R): higher power of Ricci scalar
Gauss-Bonnet: Second order contractions of Riemann with second order EoM.
Lovelock: Higher order contractions of Riemann with second order EoM.
Scalar-tensor theories (e.g., Brans-Dicke)
Vector-tensor theories
Bimetric theories (two metrics: Rosen(1940) )
Massive gravity (+graviton with mass, ghost problem)
Eddington-inspired Born-Infeld gravity

23. Gauge theory of gravity
Non-Metric theories
Einstein-Cartan theory
Include torsion tensor
Spin tensor
Metric-Affine gravitation theory
Generalization of Einstein Cartan theory
Gauge theory of gravity